# Frequentist linear mixed model

### Description

Maximum likelihood estimates for the parameters of a linear mixed model.

### Usage

1 |

### Arguments

`fixed` |
Formula describing the fixed effects part of the model, e.g. y ~ a + b + c ... If the model does not include any covariate simply state the response variable with an intercept, i.e. y ~ 1. |

`random` |
A character value with name of the column containing labels for the random effects. |

`weights` |
A numeric vector with residual weights. If not supplied, all residual weights are set to 1. |

`env.eff` |
A logical value indicating if permanent environmental effects should be included (default=FALSE). |

`data` |
A dataframe containing the data. |

`K` |
A covariance matrix for random effects. |

`verbose` |
A logical value specfying whether log messages should be printed (default = TRUE). |

### Details

The function uses a frequentist framework to fit the following linear mixed model:

*\mathbf{y} = \mathbf{Xb} + \mathbf{Zu} + \mathbf{Zp} + \mathbf{e}*

where *\mathbf{X}* is a matrix relating *\mathbf{y}* to the vector of fixed effects *\mathbf{b}*, *\mathbf{Z}* is an incidence matrix relating *\mathbf{y}* to random effects *\mathbf{u}* and *\mathbf{p}*, and *\mathbf{e}* is the vector of residuals. The likelihood of the data is assumed:

*\mathbf{y} \mid \mathbf{b},\mathbf{u},\mathbf{p},σ_{u}^{2},σ_{p}^{2},σ_{e}^2 \sim N(\mathbf{Xb},\mathbf{V})*

where *\mathbf{V} = \mathbf{ZKZ}'σ_{u}^2 + \mathbf{ZZ}'σ_{p}^2 + \mathbf{W}σ_{e}^2*, *\mathbf{K}* is a covariance matrix for *\mathbf{u}*, *σ_{u}^{2}* and *σ_{p}^{2}* are the variances of *\mathbf{u}* and *\mathbf{p}*, respectively, *\mathbf{W}* is a residual covariance matrix and *σ_{e}^{2}* is the residual variance. The current implementation assumes *\mathbf{W} = diag(w_i)*. More details about the maximization algorithm can be found in our vignette.

### Value

The returned GHap.blmm object is a list with the following items:

`b` |
A numeric vector containing the BLUE of fixed effects. |

`u` |
A numeric vector containing BLUP of correlated random effects. |

`p` |
A numeric vector containing the BLUP of permanent environmental effects. This vector is suppressed if env.eff=FALSE. |

`varu` |
A numeric value for the maximum likelihood estimate of the variance of correlated random effects. |

`varp` |
A numeric value for the maximum likelihood estimate of the variance of permanent environmental effects. This value is suppressed if env.eff=FALSE. |

`vare` |
A numeric value for the maximum likelihood estimate of the variance of residual variance. |

`h2` |
A numeric value for the maximum likelihood estimate of the variance explained by correlated random effects only. |

`H2` |
A numeric value for the maximum likelihood estimate of the variance explained by random effects. This value is suppressed if env.eff=FALSE. |

`k` |
A numeric vector containing the solutions for |

`y` |
A numeric vector containing the records used to fit the model. |

`weights` |
A numeric vector containing the residual weights used to fit the model. |

`residuals` |
A numeric vector containing residuals computed based on the BLUE and BLUP solutions. |

`pdev` |
Deviance evaluated at the BLUE and BLUP solutions. |

### Author(s)

Yuri Tani Utsunomiya <ytutsunomiya@gmail.com>

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 | ```
# #### DO NOT RUN IF NOT NECESSARY ###
#
# # Copy the example data in the current working directory
# ghap.makefile()
#
# # Load data
# phase <- ghap.loadphase("human.samples", "human.markers", "human.phase")
#
# # Subset data - randomly select 3000 markers with maf > 0.02
# maf <- ghap.maf(phase, ncores = 2)
# set.seed(1988)
# markers <- sample(phase$marker[maf > 0.02], 3000, replace = FALSE)
# phase <- ghap.subsetphase(phase, unique(phase$id), markers)
# rm(maf,markers)
#
# # Generate block coordinates based on windows of 10 markers, sliding 5 marker at a time
# blocks <- ghap.blockgen(phase, 10, 5, "marker")
#
# # Generate matrix of haplotype genotypes
# ghap.haplotyping(phase, blocks, batchsize = 100, ncores = 2, freq = 0.05, outfile = "example")
#
# # Load haplotype genotypes
# haplo <- ghap.loadhaplo("example.hapsamples", "example.hapalleles", "example.hapgenotypes")
#
# # Compute kinship matrix
# K <- ghap.kinship(haplo, batchsize = 100)
#
# # Quantitative trait with 50% heritability
# # One major haplotype accounting for 30% of the genetic variance
# sim <- ghap.simpheno(haplo = haplo, K = K, h2 = 0.5, g2 = 0.3, major = 1000,seed=1988)
#
#
# ### RUN ###
#
# #Continuous model
# model <- ghap.lmm(fixed = phenotype ~ 1, random = "individual", data = sim$data, K = K)
# model$h2
# plot(model$u,sim$u, ylab="True Breeding Value", xlab="Estimated Breeding Value")
# cor(model$u,sim$u)
``` |

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